2. Vocabulary
 Quadratic Equation – an equation that can be written
in the standard form ax 2 + bx + c = 0 where a ¹ 0
 Zero(s) of a function – x value(s) for which y = 0
* Zero(s) of a polynomial function and root(s) of a
polynomial are the same!

3. Solve by Graphing (vs. Factoring)
Recall Solve by Factoring: x 2 - 6x + 5 = 0
(x -1)(x - 5) = 0
x =1, x = 5
Solve by Graphing:

4. Example 1 Solve a quadratic equation having two solutions
Solve x2 – 2x = 3 by graphing.
SOLUTION
STEP 1 Write the equation in standard form.
x2 – 2x = 3 Write original equation.
x2 – 2x – 3 = 0 Subtract 3 from each side.
STEP 2 Graph the related function
y = x2 – 2x – 3 . The x-intercepts
are –1 and 3.

5. Example 1 Solve a quadratic equation having two solutions
ANSWER
The solutions of the equation x2 – 2x = 3 are – 1 and 3.
CHECK You can check –1 and 3 in the original equation.
x2 – 2x = 3 x2 – 2x = 3 Write original equation.
?
( – 1)2 – 2 (– 1) = 3 ?
( 3)2 – 2( 3) = 3 Substitute for x.
3 = 3 3 = 3 Simplify. Each solution
checks.

6. Example 2 Solve a quadratic equation having one solution
Solve – x2 + 2x = 1 by graphing.
SOLUTION
STEP 1 Write the equation in standard form.
– x2 + 2x = 1 Write original equation.
– x2 + 2x – 1 = 0 Subtract 1 from each side.
STEP 2 Graph the related function y = – x2 + 2x – 1 .
The x-intercept is 1.

7. Example 2 Solve a quadratic equation having one solution
ANSWER
The solution of the equation – x2 + 2x = 1 is 1.

8. Example 3 Solve a quadratic equation having no solution
Solve x2 + 7 = 4x by graphing.
SOLUTION
STEP 1 Write the equation in standard form.
x2 + 7 = 4x Write original equation.
x2 – 4x + 7 = 0 Subtract 4x from each side.
STEP 2 Graph the related function
y = x2 – 4x + 7. The graph has
no x-intercepts.

9. Example 3 Solve a quadratic equation having no solution
ANSWER
The equation x2 + 7 = 4x has no solution.

10. Number of Solutions of a Quadratic Equation
Two Solutions One Solution No Solution
A quadratic equation A quadratic equation A quadratic equation
has two solutions if has one solution if the has no real solution if
the graph of its related graph of its related the graph of its related
function has two x- function has one x- function has no x-
intercepts. intercept. intercepts.

11. Example 4 Multiple Choice Practice
The graph of the equation
y = x2 + 6x – 7 is shown. For
what value or values of x is y = 0?
x = –7 only x = 1 only
x = – 7 and x = 1 x = –1 and x = 7
SOLUTION
You can see from the graph that the x-intercepts are –7
and 1. So, y = 0 when x = –7 and x = 1.
ANSWER The correct answer is C.

12. Example 5 Approximate the zeros of a quadratic function
Approximate the zeros of y = x2 + 4x + 1 to the nearest
tenth.
SOLUTION
STEP 1 Graph the function y = x2 + 4x + 1. There are
two x-intercepts: one between – 4 and –3 and
another between –1 and 0.

13. Example 5 Approximate the zeros of a quadratic function
STEP 2 Make a table of values for x-values between
– 4 and – 3 and between – 1 and 0 using an
increment of 0.1. Look for a change in the
signs of the function values.
x – 3.9 – 3.8 – 3.7 – 3.6 – 3.5 – 3.4 – 3.3 – 3.2 – 3.1
y 0.61 0.24 – 0.11 – 0.44 – 0.75 – 1.04 – 1.31 – 1.56 – 1.79
x – 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1
y – 1.79 – 1.56 – 1.31 – 1.04 – 0.75 – 0.44 – 0.11 0.24 0.61

14. Example 5 Approximate the zeros of a quadratic function
ANSWER
In each table, the function value closest to 0 is – 0.11.
So, the zeros of y = x2 + 4x + 1 are about – 3.7 and
about – 0.3.

15. Example 6 Solve a multi-step problem
SPORTS
An athlete throws a shot put with
an initial vertical velocity of 40 feet
per second.
a. Write an equation that models
the height h (in feet) of the
shot put as a function of the
time t (in seconds) after it is
thrown.
b. Use the equation to find the time that the shot put is
in the air.

16. Example 6 Solve a multi-step problem
SOLUTION
a. Use the initial vertical velocity and height to write a
vertical motion model.
h = – 16t2 + vt + s Vertical motion model
Substitute 40 for v and
h = – 16t2 + 40t + 6.5 6.5 for s.
b. The shot put lands when h = 0. To find the time t
when h = 0, solve 0 = –16t 2 + 40t + 6.5 for t.
To solve the equation, graph the
related function h = –16t2 + 40t + 6.5
on a graphing calculator. Use the
trace feature to find the t-intercepts.

17. Example 6 Solve a multi-step problem
ANSWER
There is only one positive t-intercept. The shot put is in
the air for about 2.6 seconds.

18. Relating Roots of Polynomials,
Solutions of Equations, x-intercepts
of Graphs, and Zeros of Functions
 The Roots of the Polynomial -x 2 +8x -12 are 2 & 6.
 The Solutions of the equation -x 2 +8x -12 = 0 are
2 & 6.
 The x-intercepts of the graph of y = -x +8x -12 occur
2
where y=0, so the x-intercepts are 2 and 6.
 The Zeros of the function y = -x +8x -12 are the
2
values of x for which y=0, so the zeros are 2 and 6.

19. 10.4 Warm-Up
Solve the equation by graphing.
1. x 2 + 5x + 6 = 0
2. x 2 +8x +16 = 0
3. x 2 - 2x + 3 = 0